Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
| direct product | C4×#872 |
| Fitting | C3×C43 ⋊ C2 |
| Sylow | C3 ⋊ C4×#117 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D3 |
| commutator | C2×C3 | C22×C42 |
| Frattini | C23 | C23×D3 |
| Fitting | C3×C43 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C23×D3 | C23 |
| hypercenter | C43 | D3 |
| hypocenter | C3 | C4×#117 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C42 ◁ C43 | |
752 conjugacy classes of subgroups; only the 357 normal subgroups are shown.
This group has 752 conjugacy classes of subgroups — too many to draw. Only its 357 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(8,9,10,11)(13,17)(16,19);(12,14,15,18)(13,16,17,19);(2,3)(12,13)(14,16)(15,17)(18,19);(4,5,6,7)144 conjugacy classes — view the full character table.
| maximal subgroup of | #10684624, #10684630, #10684633, … |
| maximal normal subgroup of | #10684624, #10684630, #10684633, … |
| maximal quotient of | C4×#4932, C42×(C2×D3)⋊C4, C42×Dic3⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5872);